Balanced Ternary - Fractional Balanced Ternary

Fractional Balanced Ternary

Balanced ternary can be extended to fractional numbers similarly to how decimal numbers are written past the decimal point. For instance, is 0.1 and is 1.T.

Donald Knuth has pointed out that truncation and rounding are the same operation in balanced ternary — they produce exactly the same result(a property shared with other balanced numeral system). 0.5₁₀ is not an exception too. the 0.5₁₀ represented by the repeating fraction 0.1(round to 0, and truncate to 0) and 1.T(round to 1, and truncate to 1).

Unlike in standard base notation, where integer values and terminating fractions have multiple representations (e.g. in decimal, = 0.25₁₀ = 0.249₁₀), in balanced ternary such numbers have only one representation ( = 0.1T_3(bal)). On the other hand, numbers half of a terminating fraction (i.e. whose denominator is 2 times a power of 3) do have multiple representations e.g. = 0.01_3(bal) = 0.1T_3(bal).

The basic operations—addition, subtraction, multiplication, and division—are done as in regular ternary, except that there are some special considerations on the size compare. Multiply 2 by 10 in ternary and then divide the result by 2 to see the issue. The intermediate results are the same (in reverse order) in the two cases, and so you can use one as a check on the other.

Digit shifts multiply or divide by three instead of by two as in binary.

Multiplication by two can be done by adding a number to itself. Division by two can be done with the same operation count as an add by LeRoy Eide's algorithm (an algorithm that returns the result least significant digit first, instead of most significant digit first as standard division).